Variational integrators for perturbed non-canonical Hamiltonian systems

TL;DR

This paper introduces a coordinate-independent variational integrator for perturbed non-canonical Hamiltonian systems that efficiently leverages unperturbed dynamics for accurate long-term simulations.

Contribution

It develops a novel phase-space variational integrator applicable to perturbed non-canonical Hamiltonian systems on manifolds, with arbitrary accuracy in the perturbation parameter.

Findings

Allows $O(1)$ time steps relative to perturbation size

Coordinate independence ensures correct transformation under coordinate changes

Leverages unperturbed dynamics for improved accuracy

Abstract

Finite-dimensional non-canonical Hamiltonian systems arise naturally from Hamilton's principle in phase space. We present a method for deriving variational integrators that can be applied to perturbed non-canonical Hamiltonian systems on manifolds based on discretizing this phase-space variational principle. Relative to the perturbation parameter $\epsilon$, this type of integrator can take $O(1)$ time steps with arbitrary accuracy in $\epsilon$ by leveraging the unperturbed dynamics. Moreover, these integrators are coordinate independent in the sense that their time-advance rules transform correctly when passing from one phase space coordinate system to another.